Using logarithms with GPs

Solving exponential inequations from GP problems by taking logarithms, including the reversal of the inequality when dividing by the log of a number less than 1.

The method

Example 1●●

Which term of $5, 10, 20, \dots$ is the first to exceed $10\,000$?

$5 \times 2^{n-1} > 10\,000$, so $2^{n-1} > 2000$.

$n - 1 > \dfrac{\ln 2000}{\ln 2} \approx 10.97$, so $n > 11.97$ and $n = 12$.

Example 2: inflation●●

Prices rise by $3\%$ a year. After how many whole years will a $\$100$ item first cost more than $\$200$?

$100 \times 1.03^n > 200$, so $1.03^n > 2$.

$n > \dfrac{\ln 2}{\ln 1.03} \approx 23.4$, so after $24$ years.

Example 3: a sum●●

Find the smallest $n$ for which $1 + 1.1 + 1.1^2 + \dots + 1.1^{n-1} > 50$.

$\dfrac{1.1^n - 1}{0.1} > 50$, so $1.1^n > 6$.

$n > \dfrac{\ln 6}{\ln 1.1} \approx 18.8$, so $n = 19$.

Example 4: a ratio below 1●●●

Find the smallest $n$ with $0.9^n < 0.01$.

$n\ln 0.9 < \ln 0.01$. Since $\ln 0.9 < 0$, dividing reverses the sign: $n > \dfrac{\ln 0.01}{\ln 0.9} \approx 43.7$.

$n = 44$.

Flashcards

Solve $r^n > k$ with $r > 1$

$n > \dfrac{\ln k}{\ln r}$

Dividing by $ln r$ when $r < 1$

Reverse the inequality.

Rounding

Take the next whole number that satisfies the condition.

Practice

Which term of $5, 10, 20, \dots$ is the first to exceed $10\,000$?
Prices rise by $3\%$ a year. After how many years does a $\$100$ item first cost more than $\$200$?
Find the smallest $n$ with $1 + 1.1 + \dots + 1.1^{n-1} > 50$.
Find the smallest $n$ with $0.9^n < 0.01$.
At $5\%$ compound interest, after how many whole years does an investment first double?
Solving $0.8^n < 0.1$ by dividing by $\ln 0.8$ gives:
$n > 6.2$ and $n$ is a whole number. The answer is:
Which term of $2, 6, 18, \dots$ is the first to exceed $1000$?
Why are logarithms needed in these problems?